Podcast
Questions and Answers
Which of the following is a type of differential equation listed by Newton in his 1671 work Methodus fluxionum et Serierum Infinitarum?
Which of the following is a type of differential equation listed by Newton in his 1671 work Methodus fluxionum et Serierum Infinitarum?
- 𝑥₁∂𝑦/∂𝑥₁ + 𝑥₂∂𝑦/∂𝑥₂ = 𝑦 (correct)
- 𝑦′ + 𝑃(𝑥)𝑦 = 𝑄(𝑥)𝑦^n
- 𝑦′′ + 𝑃(𝑥)𝑦 = 𝑄(𝑥)
- 𝑑𝑦/𝑑𝑥 = 𝑓(𝑥) (correct)
Who proposed the Bernoulli differential equation in 1695?
Who proposed the Bernoulli differential equation in 1695?
- Jacob Bernoulli (correct)
- Gottfried Wilhelm Leibniz
- Isaac Newton
- Leonhard Euler
What problem was the Euler-Lagrange equation developed to solve?
What problem was the Euler-Lagrange equation developed to solve?
- The infinite series problem
- The heat flow problem
- The vibrating string problem
- The tautochrone problem (correct)
Who solved the tautochrone problem in 1755?
Who solved the tautochrone problem in 1755?
What is the name of the book in which Fourier published his work on heat flow in 1822?
What is the name of the book in which Fourier published his work on heat flow in 1822?
What is the name of the equation proposed by Fourier for conductive diffusion of heat?
What is the name of the equation proposed by Fourier for conductive diffusion of heat?
Who discovered the three-dimensional wave equation within ten years of d'Alembert's discovery of the one-dimensional wave equation?
Who discovered the three-dimensional wave equation within ten years of d'Alembert's discovery of the one-dimensional wave equation?
What is the name of the law that Fourier based his reasoning on in his work on heat flow?
What is the name of the law that Fourier based his reasoning on in his work on heat flow?
What is the primary focus of the study of differential equations?
What is the primary focus of the study of differential equations?
What do the derivatives in a differential equation typically represent?
What do the derivatives in a differential equation typically represent?
What is the significance of differential equations in various disciplines?
What is the significance of differential equations in various disciplines?
What is the limitation of solving differential equations using explicit formulas?
What is the limitation of solving differential equations using explicit formulas?
What is the purpose of numerical methods in solving differential equations?
What is the purpose of numerical methods in solving differential equations?
What is the focus of the theory of dynamical systems in relation to differential equations?
What is the focus of the theory of dynamical systems in relation to differential equations?
Who are credited with the invention of calculus, which led to the development of differential equations?
Who are credited with the invention of calculus, which led to the development of differential equations?
What is the relationship between the functions and their derivatives in a differential equation?
What is the relationship between the functions and their derivatives in a differential equation?
Flashcards are hidden until you start studying
Study Notes
Differential Equations
- A differential equation is an equation that relates one or more unknown functions and their derivatives.
- In applications, the functions represent physical quantities, and the derivatives represent their rates of change.
- Differential equations are common in many disciplines, including engineering, physics, economics, and biology.
History
- Differential equations came into existence with the invention of calculus by Isaac Newton and Gottfried Leibniz.
- In 1671, Newton listed three kinds of differential equations in his work "Methodus fluxionum et Serierum Infinitarum".
- Jacob Bernoulli proposed the Bernoulli differential equation in 1695.
- In the 1700s, Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange studied the problem of a vibrating string.
- d'Alembert discovered the one-dimensional wave equation in 1746, and Euler discovered the three-dimensional wave equation within the next ten years.
Key Developments
- The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem.
- Lagrange solved the tautochrone problem in 1755 and sent the solution to Euler.
- Both Euler and Lagrange further developed Lagrange's method and applied it to mechanics, leading to the formulation of Lagrangian mechanics.
- In 1822, Fourier published his work on heat flow, in which he based his reasoning on Newton's law of cooling.
- Fourier proposed his heat equation for conductive diffusion of heat, which is now a common part of many mathematical models.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
